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I apologise in advance if this is not the subreddit for such questions. I asked this question 24 days ago here. Despite the many answers there, I just wanted to confirm if what I understood was correct. Does the impossibility of solving the halting problem depend on the validity of the Church-Turing thesis?
I initially asked this over at /r/askcomputerscience, but there weren't any replies.

Without spending a lot of time looking into it, my immediate response is "no". The Halting Problem is independent of the mechanics of the underlying computational model - it can be stated in terms of LC without reference to TMs, or vice versa (as Turing originally did, I believe).

cypherx's response is also relevant: the HP is expressed in terms of a particular computational model - it requires being able to compose machines (terms) and reason about their behaviour. The fact that it's common to express the HP in terms of TM, then use Church-Turing to show that it can be embedded in LC, is mostly due to history. But it seems to me that the HP can be equally stated in pure LC, Post machines, universal CA, etc without any reference to other computational models.

In theory of computation, we often talk about "oracle machines" that have more power than an ordinary Turing machine. My understanding is that the Church-Turing thesis basically states that these "oracle machines" cannot be physically implemented. In other words, a Turing machine is as powerful as the most powerful computational model.

We know the halting problem is proved to be undecidable for Turing machines (no Turing machine can solve the halting problem for other Turing machines in general). I don't think anyone has proved that some more powerful machine couldn't solve the Halting problem for Turing machines. The Church-Turing thesis states that such a "more powerful machine" cannot exist, so it seems that the answer to OP's question is actually yes.

Thanks - you've clarified what OP is getting at and I think I missed. I'll try and state it in terms that make it more obvious to me, since I had to re-read your comment a few times to properly grasp it:

HP states that an Oracle for <formal system> cannot be realised in <formal system>. It leaves open the possibility for a stronger formal system to exist that can construct an oracle. CT claims that there exist no stronger formal systems than Turing Machines: thus, Oracles cannot exist at all.

So the answer to OP's question is yes, if you take a wider view of "the impossibility of solving the halting problem" than I did. My narrower worldview missed the point :-).

I don't think anyone has proved that some more powerful machine couldn't solve the Halting problem for Turing machines.

You can easily construct an oracle machine that can solve the halting problem: simply take the Turing machine and add an oracle for the halting problem for Turing machines! You can construct a whole hierarchy this way: take level 0 to be the Turing machine, and level n>0 machine is the level n-1 machine with an oracle that can solve the halting problem for level n-1 machines. No oracle machine can solve its own halting problem, so all the levels of the hierarchy are different.

When people say "the halting problem", what they mean is usually the halting problem for Turing machines, though :) Even more concretely, they are probably thinking about solving the halting problem for actual, real-world software. For that, the halting problem for Turing machines is the most appropriate model (even though all real-world computers are finite state machines, the set of states is not quite finite enough in practice - maybe the ultrafinitists are onto something ;).

You are correct, Alpha_Q, the unsolvability of the Halting Problem is dependent on the truth of the Church-Turing Thesis, as follows "Every effectively calculable function is a computable function", where "computable" means computable via Turing Machine, and "effectively calculable" refers to the functions that can, by any method, be algorithmically (i.e. mechanically) instantiated. While Turing Machines are creatures of logic, the instantiation criterion brings the CTT within the realm of constructive mathematics, and thus of physics, because any algorithm constructed as such would have an isomorphic physical counterpart (e.g. a set of logic gates). The Physical Church-Turing Thesis makes this explicit, and strengthens the claim to "Any physically realizable calculable function can be computed via Turing Machine (or equivalent)".

There are, however, other logics (with, for example, transfinite-induction) that can solve the Halting Problem as formulated for Turing Machines, because they allow for the construction of oracle machines with continuous as opposed to discrete tapes (and thus the Diagonalization Argument which proves the Halting Problem incomputable fails, because the Continuous-Tape Machine has real, aleph-one precision).

This then opens an avenue to the realization of super-Turing computation in the physical world, as it is yet an open question as to whether the manipulable substrate of the Universe is discrete (quantum mechanics, chromodynamics, etc. would say so) or continuous (general relativity would oppose). By implicitly making claims of two different fields, mathematical logic and physics, the Church-Turing Thesis ascends further, into the realm of the meta-physical, and demonstrates the integrated nature underlying the foundations of, at least, these two disciplines. A definitive statement from either side would have equally manifest repercussions for the other.

I don't agree. First, what is your formal definition of "effectively calculable"? It seems like it is an informal concept, which means that the Church-Turing Thesis is not a formal statement, but an informal statement making the bridge with a formal theory. Why would, then, a formal theorem (the halting problem is unsolvable), depend on an informal statement?

What is you definition of the unsolvability of the halting problem? If formulated as "There is no turing machine that decides whether a given machine halts on a given input"¹, I don't think it depends on the Church-Turing thesis. Do you have an alternative formulation? What is the interest of this different formulation, does it tell us anything more interesting, or does it just bundle the Church-Turing thesis inside the theorem?

¹: this formalizations have been shown equivalent to "in the untyped lambda-calculus, there is no provably normalizing lambda-term that takes (an encoding) a lambda-term as argument and decides if it is terminating".

It is not formally defined. That's the point of Turing machines: to give a formal definition ("Turing computable") for this intuitively obvious concept ("effectively computable"). Most people agree that Turing computability cinches it. The Church-Turing thesis is simply the claim that it does so (i.e. that Turing computable = effectively computable).

It seems like it is an informal concept, which means that the Church-Turing Thesis is not a formal statement, but an informal statement making the bridge with a formal theory.

It is not a formal statement.

The physical Church-Turing thesis is less vague than the basic one: it claims that the Universe is Turing-computable. That's not formal, either, though. You'd need a formal Theory of Everything and then show that it is simulatable by a Turing machine to make it formal, but then it wouldn't be the physical Church-Turing thesis anymore, as you can never know that the TOE is correct.

Why would, then, a formal theorem (the halting problem is unsolvable), depend on an informal statement?

It is not necessarily a formal statement! If you mean it as "the halting problem is unsolvable by a Turing machine", then yes, it's a theorem. If you mean it as "the halting problem is unsolvable by any device that can be physically constructed", then it isn't, and can never be (maybe physicists will one day discover the Oracle Particle with God's phone number written on it :). Theorems are always about models of reality, never about reality itself. I think the OP meant to ask about reality, in which case yes, there is a dependency on the physical Church-Turing thesis.

What is the interest of this different formulation

It's where the rubber hits the road: few people are interested in mathematics as a pointless game of symbols played according to formal rules; interest arises only when the game is somehow (by a version of a Church-Turing thesis) claimed to have some real-world implications.

Coffee2theorems responded well to most of your points, so I would like only to address one critical element which was outside the focus of your comment: the equivalence between constructivist mathematics and physically constructable logic gates, and the resulting interplay of fundamental questions in computation and physics. Because Turing-esque machines can in fact be built and operated within physical reality (these being computers, which are instantiations of Turing-complete machines modulo a finite, but ever lengthening due to technological advances, tape), the Church-Turing Thesis implicitly makes a statement about the physical extent of algorithmic mathematics, like so: "Every effectively calculable function is a Turing computable function" -> "Every Turing computable function can, in theory and in fact, be translated into a series of activations on a particular set of logic gates, made of e.g. silicon or billiard balls" -> "Effectively calculable functions comprise those whose operation is physically effectively realizable, given certain assumptions about the physical nature".

Thus there exists an overlap and an integration between the material and logical domains, as the particular computational limits of physical computers described by discrete mathematical logic, formalized in the model of a Turing Machine or its equivalent ilk, are susceptible to corroboration or countermanding by discoveries concerning the divisibility of space-time, and super-Turing formalisms alike await the possibility of empirical investigation so long confined to the Turing-esque population.

Is there a general-case halting problem which we can talk about without reference to a particular computational model? When you say "Halting Problem" aren't you already implicitly talking about some particular variant (i.e. "will this TM halt?" "will this lambda term normalize?" etc...)?

We don't need to frame it any more generally than that, since the entire point of the Church-Turing thesis is that all sufficiently powerful models of computation are equivalent with respect to the set of functions they can compute. Any other model of computation you wish to consider could be simulated on a Turing machine.

Which is great, until we start talking about whether the Halting Problem "depends" on the equivalence of all known computational models. At that point, I'd like to know which Halting Problem you're talking about.

The theorem is that HP is unsolvable on a TM. (Corollaries of that are that HP is unsolvable on a register machine, unsolvable in lambda calculus, etc.) So in that sense, the theorem does not depend on Church's Thesis.

But the theorem takes on great significance with the (very convincing) evidence for CT. If CT is so then the fact that HP is not solvable on a TM means that HP is not solvable by us.

The Church-Turing thesis is not a theorem that can be proven to be valid. It is just a thesis which can be accepted or rejected. The halting problem is not dependent on whether the church-turing thesis is accepted, because it depends on the mechanism which it is defined for.